Hello!!!

Which relation do the constants $a,b$ have to satisfy so that the implicit function theorem implies that the system of two equations

$$axu^2v+byv^2=-a \ \ \ \ bxyu-auv^2=-a$$

can be solved as for u and v as functions $u=u(x,y)$ and $v=v(x,y)$ with continuous partial derivatives of first order in some region of $(1,0)$ and with u(1,0)=1, v(1,0)=-1.

I have thought the following:

$$\Delta=\begin{pmatrix}

au^2v & bv^2\\

byu & -bxu

\end{pmatrix}=\begin{pmatrix}

-a & b\\

0 & -b

\end{pmatrix}$$

It should hold that $det(\Delta)\neq0 \Rightarrow ab \neq 0$.

So the condition is $a^2+b^2 \neq 0$. Am I right?